5.1 Normal Distribution and its Properties
2 min read•july 23, 2024
Normal distributions are the backbone of many statistical analyses. They're characterized by their symmetric bell shape and defined by two key parameters: the and . These distributions are incredibly useful for modeling real-world phenomena and making predictions.
Understanding normal distributions is crucial for probability calculations. The helps estimate probabilities within certain ranges, while z-scores standardize values for easier comparison. Probability density and cumulative distribution functions further aid in calculating specific probabilities for various scenarios.
Properties and Characteristics of Normal Distributions
Properties of normal distribution
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- Symmetric bell-shaped curve represents the shape of the distribution
- Mean, median, and mode are equal located at the center (peak) of the distribution
- Continuous probability distribution represents the data
- Probability represented by the area under the curve (total area always equals 1)
- Extends infinitely in both directions (left and right)
- Asymptotically approaches the x-axis but never touches it (tails extend indefinitely)
- Defined by two parameters: mean () and standard deviation ()
- (empirical rule) applies to normal distributions
- Specifies the percentage of data within 1, 2, and 3 standard deviations of the mean
Parameters of normal distribution
- Mean () determines the location (center) of the distribution
- Shifting the mean shifts the entire distribution left (negative shift) or right (positive shift)
- Standard deviation () determines the spread (width) of the distribution
- Larger standard deviation results in a wider, flatter distribution (more variability)
- Smaller standard deviation results in a narrower, taller distribution (less variability)
Probability Calculations and Functions
Probabilities using empirical rule
- Empirical rule (68-95-99.7 rule) specifies data percentages within standard deviations
- Approximately 68% of data falls within one standard deviation of the mean ()
- Approximately 95% of data falls within two standard deviations of the mean ()
- Approximately 99.7% of data falls within three standard deviations of the mean ()
- Z-scores standardize values to represent the number of standard deviations an observation is from the mean
- Formula: (observation minus mean, divided by standard deviation)
- Can be used to calculate probabilities using a ()
Density vs cumulative distribution functions
- Probability density function (PDF) denoted as
- Describes the relative likelihood of a random variable taking on a specific value
- Area under the curve between two points represents the probability of the random variable falling within that range (not the height of the curve)
- Cumulative distribution function (CDF) denoted as
- Describes the probability that a random variable is less than or equal to a specific value
- Monotonically increasing function, ranging from 0 to 1 (never decreases)
- , where is the random variable
- Can be used to calculate probabilities for intervals by subtracting CDF values (upper bound minus lower bound)
Key Terms to Review (17)
68-95-99.7 Rule: The 68-95-99.7 rule, also known as the empirical rule, describes how data is distributed in a normal distribution. Specifically, it states that approximately 68% of the data falls within one standard deviation from the mean, about 95% within two standard deviations, and around 99.7% within three standard deviations. This rule highlights the predictable nature of data in a normal distribution and is essential for understanding variability and making inferences.
Bell curve: A bell curve is a graphical representation of a normal distribution, characterized by its symmetric, bell-shaped appearance. This shape indicates that data points are more concentrated around the mean, with fewer occurrences as you move away from the center in either direction. It highlights the properties of normal distribution, including the empirical rule which states that approximately 68% of data falls within one standard deviation of the mean.
Central Limit Theorem: The Central Limit Theorem states that, given a sufficiently large sample size, the sampling distribution of the sample mean (or sample proportion) will be normally distributed, regardless of the original population's distribution. This theorem is crucial because it allows for making inferences about population parameters using sample statistics, bridging the gap between descriptive statistics and inferential statistics.
Confidence Interval: A confidence interval is a range of values that is used to estimate an unknown population parameter, calculated from sample data. It provides an interval within which we expect the true parameter to fall with a certain level of confidence, typically expressed as a percentage like 95% or 99%. This concept is fundamental in statistical inference, allowing us to make conclusions about populations based on sample data.
Empirical Rule: The empirical rule, often referred to as the 68-95-99.7 rule, states that for a normal distribution, approximately 68% of the data points will fall within one standard deviation of the mean, about 95% within two standard deviations, and nearly all (99.7%) within three standard deviations. This rule is fundamental for understanding the spread and behavior of data in a normal distribution and provides a quick way to assess probabilities.
Histogram: A histogram is a graphical representation of the distribution of numerical data that uses bars to show the frequency of data points within specified ranges, known as bins. It provides a visual interpretation of data that helps to identify patterns such as central tendency, dispersion, and the shape of the distribution, making it a fundamental tool in understanding data characteristics.
Hypothesis Testing: Hypothesis testing is a statistical method used to make inferences about a population based on sample data. It involves formulating two competing statements, the null hypothesis and the alternative hypothesis, and using sample data to determine which statement is supported by the evidence. This process is crucial for decision-making and helps to assess the validity of claims in various contexts, particularly in business and research.
Mean: The mean, often referred to as the average, is a measure of central tendency that is calculated by summing all values in a dataset and dividing by the total number of values. This concept is crucial for making informed decisions based on data analysis, as it provides a single value that represents the overall trend in a dataset.
N(μ, σ²): The notation n(μ, σ²) represents a normal distribution characterized by its mean (μ) and variance (σ²). This notation is crucial in statistics as it defines the shape and location of the normal curve, which is symmetric around the mean. The normal distribution is widely used in business statistics for modeling real-world phenomena, making this term essential for understanding probability in various contexts.
Normal curve: The normal curve, also known as the Gaussian curve, is a symmetric, bell-shaped graph that represents the distribution of a continuous random variable. This curve is characterized by its mean and standard deviation, where the highest point corresponds to the mean and indicates the center of the data distribution. The shape of the normal curve is essential in probability and statistics as it describes how data points are spread around the mean, allowing for various statistical analyses.
Normal distribution: Normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. This characteristic forms a bell-shaped curve, which is significant in various statistical methods and analyses.
Percentiles: Percentiles are values that divide a dataset into 100 equal parts, indicating the relative standing of a value within that dataset. This concept helps in understanding how a particular score compares to others, making it easier to interpret data distributions. Percentiles are commonly used in statistics to summarize information and provide insight into the distribution's characteristics, such as its spread and central tendency.
Sampling distribution: A sampling distribution is the probability distribution of a statistic obtained by selecting random samples from a population. It reflects the variability of the statistic, such as the mean or proportion, across different samples, and is crucial for making inferences about the population based on sample data. Understanding sampling distributions helps in assessing how sample statistics behave, particularly when considering larger samples and the application of various statistical methods.
Standard Deviation: Standard deviation is a statistical measure that quantifies the amount of variation or dispersion of a set of values. It indicates how much individual data points deviate from the mean (average) of the data set, helping to understand the spread and reliability of the data in business contexts.
Standard Normal Distribution: The standard normal distribution is a special case of the normal distribution where the mean is 0 and the standard deviation is 1. It serves as a reference for comparing different normal distributions and helps in determining probabilities and percentiles. By converting values from any normal distribution to this standardized form using Z-scores, one can easily interpret and analyze data across various contexts.
Symmetry: Symmetry in statistics refers to a balanced and proportionate arrangement of data points around a central value, such that the left side of the distribution mirrors the right side. This concept is crucial as it indicates that the mean, median, and mode are all located at the center of the distribution, suggesting that the data is evenly distributed. Symmetry plays a vital role in understanding normal distributions and non-parametric tests, where it affects the assumptions and results derived from statistical analyses.
Z-score: A z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It indicates how many standard deviations an element is from the mean, allowing for comparison between different datasets and understanding the relative position of a value within a distribution.